How to draw rounds. Pairings. Learning new material

Often, when depicting a part contour in a drawing, it is necessary to perform a smooth transition from one line to another (a smooth transition between straight lines or circles) to meet design and technological requirements. A smooth transition from one line to another is called conjugation.

To build conjugations, you need to define:

• interface centers(centers from which arcs are drawn);
• touch points/pairing points(points at which one line passes into another);
• fillet radius(if it is not set).

Consider the main types of conjugations.

Conjugation (tangency) of a straight line and a circle

Construction of a straight line tangent to a circle. When constructing a conjugation of a straight line and a circle, a well-known sign of tangency of these lines is used: a straight line tangent to a circle makes a right angle with a radius drawn to the tangent point (Fig. 1.12).

Rice. 1.12.

To- touch point

To draw a tangent to a circle through a point A lying outside the circle, it is necessary:

• 1) connect given point BUT(Fig. 1.13) with the center of the circle O;
• 2) cut OA halve (OS = SA, see fig. 1.7) and draw an auxiliary circle with a radius SO(or SA);

Rice. 1.13.

3) point /C, (or TO." since the problem has two solutions) connect with a dot BUT.

Line AK^(or AK.,) is tangent to the given circle. points K i and K 2 - touch points.

It should be noted that Fig. 1.13 also illustrates one of the methods for accurately graphical construction of two perpendicular lines (tangent and radius).

Construction of a straight line tangent to two circles. We draw the reader's attention to the fact that the problem of constructing a straight line tangent to two circles can be considered as a generalized case of the previous problem (constructing a tangent from a point to a circle). The similarity of these tasks can be seen from Fig. 1.13 and 1.14.

External tangency of two circles. With external tangency (see Fig. 1.14), both circles lie on the same side of the straight line.

On fig. 1.14 shows a small circle with a radius R centered on a point BUT and a great circle with a radius R( centered on

Rice. 1.14. Construction of an external tangent to two circles ke O. To construct an external tangent to these circles, you must do the following:

• 1) through the center O draw an auxiliary circle of radius (/?, - R);
• 2) construct tangents to the auxiliary circle from the point BUT(center of the small circle). points TO ( and TO.,- tangent points of lines and a circle (note that the problem has two solutions);
• 3) points TO ( and K 2 connect to the center O and continue these lines until they intersect with a circle with a radius Rv Intersection points K l and /C, are points of contact (conjugation);
• 4) through a point BUT draw radii parallel to lines ()K L and ok g The points of intersection of these radii with a small circle are points TO- and K l are points of contact (conjugation);
• 5) connecting the dots K l and /C (; , and also K l and K 5, get the required tangents.

Internal touch of two circles (the circles lie on opposite sides of the straight line, Fig. 1.15) is performed by analogy with the external touch, with the only difference being that an auxiliary circle with a radius /?, + R. Pa fig. 1.15 shows two possible solutions to the problem.

Rice. 1.1

Conjugation of intersecting lines by an arc of a circle with a given radius. The construction (Fig. 1.16) is reduced to the construction of a circle with a radius R, tangent to both given lines at the same time.

To find the center of this circle, we draw two auxiliary lines parallel to the given ones, at a distance R from each of them. The point of intersection of these lines is the center O conjugation arcs. Perpendiculars dropped from the center O on the given lines, determine the points of conjugation (tangency) /C, and K 2 .

Rice. 1.16.

Rice. 1.17. Constructing a conjugation of a circle and a straight arc with a given radius R:

a- internal touch; b- external touch

Conjugation of a circle and a straight arc with a given radius.

Examples of constructing conjugations of a circle and a straight arc with a given radius R shown in fig. 1.17.

In this short article, the main types of conjugations will be considered and you will learn how to construct a conjugation of angles, straight lines, circles and arcs, circles with a straight line.

Conjugation is called smooth transition from one line to another. In order to build a conjugation, you need to find the conjugation center and conjugation points.

Pairing point is a common point for mating lines. The junction point is also called the transition point.

Below are the main mate types.

Conjugation of angles (Conjugation of intersecting lines)

Conjugation of a right angle (Conjugation of intersecting lines at a right angle)

AT this example building will be considered conjugation right angle given junction radius R. First of all, let's find the junction points. To find the junction points, you need to put a compass at the vertex of a right angle and draw an arc with a radius R until it intersects with the sides of the angle. The resulting points will be the conjugation points. Next, you need to find the center of pairing. The center of the mate will be a point equidistant from the sides of the corner. Let's draw two arcs from points a and b with a conjugation radius R until they intersect with each other. The point O obtained at the intersection will be the center of conjugation. Now, from the center of the junction of point O, we describe the arc with the junction radius R from point a to point b. The conjugation of the right angle is built.

Conjugation of an acute angle (Conjugation of intersecting lines at an acute angle)

Another example of corner conjugation. This example will build conjugation
acute angle. To construct a conjugation of an acute angle with a compass opening equal to the conjugation radius R, we draw two arcs from two arbitrary points on each side of the angle. Then we draw tangents to the arcs until they intersect at point O, the center of conjugation. From the resulting center of conjugation, we lower the perpendicular to each of the sides of the corner. So we get the conjugation points a and b. Then we draw from the center of the pairing, point O, an arc with the radius of the pairing R, connecting the pairing points a
and b. The conjugation of an acute angle is constructed.

Conjugation of an obtuse angle (Conjugation of intersecting lines at an obtuse angle)

It is built by analogy with the conjugation of an acute angle. We also, first with a radius R, draw two arcs from two arbitrarily taken points on each side, and then draw tangents to these arcs until they intersect at point O, the center of the pair. Then we lower the perpendiculars from the center of conjugation to each of the sides and connect with an arc equal to the radius of conjugation of the obtuse angle R, the obtained points a and b.

Pairing Parallel Straight Lines

Let's build conjugation of two parallel lines. We are given a conjugation point a lying on one straight line. Draw a perpendicular from point a until it intersects with another line at point b. Points a and b are junction points of straight lines. Drawing an arc from each point, with a radius greater than the segment ab, we find the center of conjugation - point O. From the center of conjugation, we draw an arc of a given radius of conjugation R.

Conjugation of circles (arcs) with a straight line

External fillet of an arc and a straight line

In this example, a conjugation with a given radius r of a straight line given by segment AB and a circular arc with radius R will be constructed.

First, find the center of conjugation. To do this, draw a straight line parallel to the segment AB and spaced from it by a distance of the radius of conjugation r, and an arc from the center of the circle OR with radius R + r. The point of intersection of the arc and the straight line will be the center of conjugation - the point Or.

From the center of conjugation, point Or, let's drop the perpendicular to the line AB. Point D, obtained at the intersection of the perpendicular and segment AB, will be the conjugation point. Find the second point of conjugation on the arc of the circle. To do this, we connect the center of the circle OR and the center of conjugation Or with a line. Let's get the second point of conjugation - point C. From the center of conjugation, draw an arc of conjugation with radius r, connecting the conjugation points.

Internal fillet of a straight line with an arc

By analogy, an internal conjugation of a straight line with an arc is constructed. Consider an example of constructing a conjugation with radius r of a straight line defined by segment AB and a circle arc of radius R. Find the center of conjugation. To do this, we construct a straight line parallel to the segment AB and spaced from it at a distance of radius r, and an arc from the center of the circle OR with radius R-r. The point Or, obtained at the intersection of the line and the arc, will be the center of the conjugation.

From the center of conjugation (point Or) we drop the perpendicular to the line AB. Point D, obtained on the basis of the perpendicular, will be the junction point.

To find the second point of conjugation on the arc of a circle, connect the center of conjugation Or and the center of the circle OR with a straight line. At the intersection of the line with the arc of the circle, we get the second junction point - point C. From the point Or, the junction center, we draw an arc with radius r, connecting the junction points.

Conjugation of circles (arcs)

External pairing conjugation is considered, in which the centers of conjugated circles (arcs) O1 (radius R1) and O2 (radius R2) are located behind the conjugating arc of radius R. The example considers the external conjugation of arcs. First, we find the center of conjugation. The conjugation center is the point of intersection of arcs of circles with radii R+R1 and R+R2, constructed from the centers of the circles O1(R1) and O2(R2), respectively. Then we connect the centers of the circles O1 and O2 with straight lines to the center of the mate, point O, and at the intersection of the lines with the circles O1 and O2 we get the mates A and B. After that, from the center of the mate we build an arc of a given radius of the mate R and connect the points A and B to it .

Internal pairing conjugation is called, in which the centers of conjugated arcs O1, radius R1, and O2, radius R2, are located inside the arc conjugating them with a given radius R. The picture below shows an example of constructing an internal conjugation of circles (arcs). First, we find the center of conjugation, which is the point O, the point of intersection of arcs of circles with radii R-R1 and R-R2 drawn from the centers of the circles O1 and O2, respectively. Then we connect the centers of the circles O1 and O2 with straight lines with the center of conjugation and at the intersection of the lines with circles O1 and O2 we get conjugation points A and B. Then from the center of conjugation we build an arc of conjugation of radius R and build a conjugation.

Mixed conjugation of arcs is a conjugation, in which the center of one of the mating arcs (O1) lies outside the arc of radius R conjugating them, and the center of the other circle (O2) lies inside it. The illustration below shows an example of a mixed pairing of circles. First, we find the center of conjugation, point O. To find the center of conjugation, we build arcs of circles with radii R + R1, from the center of a circle of radius R1 to point O1, and R-R2, from the center of a circle of radius R2 to point O2. Then we connect the center of the pairing point O with the centers of the circles O1 and O2 straight lines and at the intersection with the lines of the corresponding circles we get the pairing points A and B. Then we build the pairing.

Drawing

Grade 9

Subject: Pairing.

Goals:

1. Educational:

Know the definition of conjugation, types of conjugations.

Be able to build conjugations and explain the construction process.

2. Developing:

Develop spatial thinking.

Create conditions for the development of cognitive interest.

3. Educational:

To promote the formation of a respectful attitude towards comrades (the ability to listen and hear).

Cultivate accuracy when making drawings.

Teaching methods:

explanatory and illustrative;

Form of organization cognitive activity:

frontal;

individual.

Lesson type:

Combined

I. During the classes

1. Organizational moment:

greetings;

checking student attendance;

filling in a classroom journal by a teacher;

readiness check.

The message of the topic and purpose of the lesson:

2. Actualization of students' knowledge:

Questions:

Tell us about the sequence of graphic images that you need to perform in order to divide the segment into several equal parts.

How to divide a circle into 2, 4 and 8 equal parts?

How to divide a circle into three, six and twelve equal parts?

3. Study of new material.

3.1. Pairings

3.2. Conjugation of two straight lines by an arc of a given radius.

3.3. Application of geometric constructions in practice.

3.1. Pairings

The template (Appendix 8) has rounded corners. Straight lines smoothly turn into curves.

The smooth transition of a straight line into a curve or a curved line into another curve is called conjugation.

To build a conjugation, you need to find the centers from which the arcs are drawn, which means the centers of conjugations. It is also necessary to find the points at which one line passes into the second, which means points of conjugation.

Thus, to build any mate, you need to find the following elements: the center of the mate, the points of the mate - and you need to know the radius of the mate

3.2. Pairingstwo straight lines with an arc of a given radius. Given straight lines that add right, acute and obtuse angles (Appendix 8.1, a), and the value of the radii of the conjugation arc.

It is necessary to construct conjugations of these lines by an arc of a given radius.

For all three cases, the general construction method is used.

1. Find point 0 - the center of pairing (Appendix 8.1, b). It must lie at a distance R from the given lines. Obviously, this condition is satisfied by the point of intersection of two lines placed parallel to the given ones at distances R from them. To draw these lines, perpendiculars are erected from arbitrarily chosen points of each given line. Set aside for their length of radius R. Through the resulting points, straight lines are drawn parallel to the given ones.

At the point of intersection of these lines is the center O of conjugation.

2. Find the interface points (Appendix 8.1, c). To do this, perpendiculars are lowered from the center of conjugation (point 0) to the given lines. The resulting points are conjugation points.

3. Putting the support leg of the compass at point 0, describe an arc of a given radius R between the junction points (Appendix 8.1, c).

Two elements: the center and junction points are obligatory when building any mates.

3.3. Applicationgeometric constructions in practice.

To make some detail from a metal sheet, for example, the template shown in (Appendix 8), you must first of all trace its contour on the metal, this means making a markup. There are many similarities between drawing and markup.

To perform a drawing or markup, you need to determine which of the geometric constructions you need to apply, which means to analyze the graphic composition of the image. On the left in (Appendix 8.2) the constructions are shown, from which the work on tracing the outline of the template is added.

As a result of the analysis, we establish that the outline of the template contour consists mainly of constructing an angle of 60° and conjugating acute and obtuse angles with arcs of given radii.

What is the pattern markup sequence? Does it need to start with the construction of conjugation? This cannot be done.

The correct sequence for constructing a drawing is shown in (Appendix 8.3).

First, those drawing lines are drawn, the position of which is determined by the given dimensions. and does not require additional constructions, and then conjugations are built. Means:

1) draw the center line and the base line of the template (Appendix 8.3, a). From the center line to the right and to the left set aside half the length of the base, this. means 50 mm;

2) build angles of 60 ° and draw a line parallel to the base at a distance of 50 mm from it (Appendix 8.3, b);

3) find the centers of mates (Appendix 8.3, c);

4) determine the points of conjugation (Fig. 143, d);

5) circle the arcs of mates. Outline the visible contour and apply dimensions (Appendix 8.3, e).

4. Physical education for the eyes.

At an average pace, do three to four circular movements with your eyes in right side, the same to the left side. After relaxing the eye muscles, look into the distance at the expense of 1-6. repeat 1-2 times.

II. Practical work

1. Introductory briefing:

In the workbook, draw the second half of the symmetrical figure (Appendix 8.4).

Perform the exercise on building connections (Appendix 8.5 1, 2, 3). The sizes are arbitrary.

2. Independent work:

3. Current briefing:

• Identification and correction of typical errors;

  control over the implementation of safety rules;

  helping students;

4. Final part.

Analysis of the performed practical work.

Grading.

Installation for the next lesson:

Homework instructions:

According to the textbook "Drafting" paragraph 1.10, exercise fig. 1.63

Workplace cleaning

Annex 8

Appendix 8.1

Appendix 8.2

Appendix 8.3 Appendix 8.4

Appendix 8.5

Pairing Center- a point equidistant from the mating lines. And the common point for these lines is called conjugation point .

The construction of conjugations is performed using a compass.

The following types of pairing are possible:

1) conjugation of intersecting lines using an arc of a given radius R (rounding corners);

2) conjugation of a circular arc and a straight line using an arc of a given radius R;

3) conjugation of arcs of circles of radii R 1 and R 2 by a straight line;

4) conjugation of arcs of two circles of radii R 1 and R 2 by an arc of a given radius R (external, internal and mixed conjugation).

With external mating, the centers of the mating arcs of radius R 1 and R 2 lie outside the mating arc of radius R. With internal mating, the centers of the mating arcs lie inside the mating arc of radius R. With mixed mating, the center of one of the mating arcs lies inside the mating arc of radius R, and the center of the other mating arc - outside it.

In table. 1 shows the construction and gives brief explanations for the construction of simple conjugations.

PairingsTable 1

An example of simple mates	Graphic construction of mates	Brief explanation for the construction
1. Conjugation of intersecting lines using an arc of a given radius R.		Draw straight lines parallel to the sides of the angle at a distance R. From a point O mutual intersection of these lines, lowering the perpendiculars to the sides of the angle, we get the conjugation points 1 and 2 . Radius R draw an arc.
2. Conjugation of a circular arc and a straight line using an arc of a given radius R.		On distance R draw a line parallel to a given line, and from the center O 1 with a radius R+R 1- an arc of a circle. Dot O- the center of the conjugation arc. Point 2 we get on the perpendicular drawn from point O to a given straight line, and point 1 - on a straight line OO 1 .
3. Conjugation of arcs of two circles of radii R1 and R2 straight line.		From point O 1 draw a circle with radius R 1 - R2. The segment O 1 O 2 is divided in half and from the point O 3 draw an arc with a radius of 0.5 O 1 O 2 . Connect points O 1 and O 2 with a point BUT. From point O 2, drop the perpendicular to the line AO 2, points 1.2 - pairing points.

Table 1 continued

4. Conjugation of arcs of two circles of radii R1 and R2 arc of a given radius R(external pairing).		From centers O 1 and O 2 draw arcs of radii R+R 1 and R + R 2 . O 1 and O 2 with point O. Points 1 and 2 are junction points.
5. Conjugation of arcs of two circles of radii R1 and R2 arc of a given radius R(internal pairing).		From centers O 1 and O 2 draw arcs of radii R-R1 and R-R2. We get a point O- the center of the conjugation arc. connect the dots O 1 and O 2 with point O until the intersection with the given circles. points 1 and 2- junction points.
6. Conjugation of arcs of two circles of radii R1 and R2 arc of a given radius R(mixed conjugation).		From the centers O 1 and O 2 draw arcs of radii R- R 1 and R + R 2 . We get point O - the center of the conjugation arc. connect the dots O 1 and O 2 with point O until the intersection with the given circles. points 1and 2- junction points.

curved curves

These are curved lines, in which the curvature continuously changes on each of their elements. Curved curves cannot be drawn with a compass, they are constructed from a series of points. When drawing a curve, the resulting series of points is connected along a pattern, so it is called a curved line. The accuracy of building a curved curve increases with an increase in the number of intermediate points on a curve section.

The curved curves include the so-called flat sections of the cone - ellipse, parabola, hyperbola, which are obtained as a result of the section of a circular cone by a plane. Such curves were considered when studying the course "Descriptive Geometry". Curves also include involute, sinusoid, spiral of Archimedes, cycloidal curves.

Ellipse- the locus of points, the sum of the distances of which to two fixed points (foci) is a constant value.

The most widely used method of constructing an ellipse along the given semiaxes AB and CD. When constructing, two concentric circles are drawn, the diameters of which are equal to the given axes of the ellipse. To build 12 points of an ellipse, the circles are divided into 12 equal parts and the resulting points are connected to the center.

On fig. 15 shows the construction of six points of the upper half of the ellipse; the lower half is drawn in the same way.

Involute- is the trajectory of a circle point formed by its deployment and straightening (circle development).

The construction of an involute according to a given diameter of a circle is shown in fig. 16. The circle is divided into eight equal parts. From points 1,2,3 draw tangents to the circle, directed in one direction. On the last tangent, the involute step is set equal to the circumference

(2 pR), and the resulting segment is also divided into 8 equal parts. Putting one part on the first tangent, two parts on the second, three parts on the third, etc., we get the involute points.

Cycloid curves- flat curved lines described by a point belonging to a circle rolling without slipping along a straight line or circle. If at the same time the circle rolls in a straight line, then the point describes a curve called a cycloid.

The construction of a cycloid according to a given circle diameter d is shown in Fig.17.

Rice. 17

A circle and a segment of length 2pR are divided into 12 equal parts. Draw a straight line through the center of the circle parallel to the line segment. From the points of division of the segment to the straight line, perpendiculars are drawn. At the points of their intersection with the straight line, we get O 1, O 2, O 3, etc. are the centers of the rolling circle.

From these centers we describe arcs of radius R. Through the division points of the circle we draw straight lines parallel to the straight line connecting the centers of the circles. At the intersection of the straight line passing through point 1 with the arc described from the center O1, there is one of the points of the cycloid; through point 2 with another from the center O2 - another point, etc.

If the circle rolls along another circle, being inside it (along the concave part), then the point describes a curve called hypocycloid. If a circle rolls along another circle, being outside it (along the convex part), then the point describes a curve called epicycloid.

The construction of a hypocycloid and an epicycloid is similar, but instead of a segment of length 2pR, an arc of the guide circle is taken.

The construction of an epicycloid according to a given radius of the movable and fixed circles is shown in Fig.18. Angle α, which is calculated by the formula

α = 180°(2r/R), and the circle of radius R is divided into eight equal parts. An arc of a circle of radius R + r is drawn and from points О 1 , О 2 , О 3 .. - a circle of radius r.

The construction of the hypocycloid by the given radii of the moving and fixed circles is shown in Fig.19. The angle α, which is calculated, and the circle of radius R are divided into eight equal parts. An arc of a circle with radius R - r is drawn and from points O 1, O 2, O 3 ... - a circle with radius r.

Parabola- this is the locus of points equidistant from a fixed point - the focus F and a fixed line - the directrix, perpendicular to the axis of symmetry of the parabola. The construction of a parabola according to a given segment OO \u003d AB and a chord CD is shown in Fig. 20

Direct OE and OS are divided into the same number of equal parts. Further construction is clear from the drawing.

Hyperbola- the locus of points, the difference in the distances of which from two fixed points (foci) - is a constant value. Represents two open, symmetrically located branches.

The constant points of the hyperbola F 1 and F 2 are foci, and the distance between them is called focal. The line segments connecting the points of the curve with the foci are called radius vectors. A hyperbola has two mutually perpendicular axes - real and imaginary. The lines passing through the center of intersection of the axes are called asymptotes.

The construction of a hyperbola according to a given focal length F 1 F 2 and the angle α between the asymptotes is shown in Fig.21. An axis is drawn on which the focal length is plotted, which is halved by point O. A circle of radius 0.5F 1 F 2 is drawn through point O until it intersects at points C, D, E, K. Connecting points C with D and E with K, one obtains points A and B are the vertices of the hyperbola. From point F 1 to the left, arbitrary points 1, 2, 3 are marked ... the distances between which should increase as they move away from the focus. From focal points F 1 and F 2 with radii R=B4 and r=A4, arcs are drawn to mutual intersection. Intersection points 4 are points of the hyperbola. The rest of the points are constructed in a similar way.

sinusoid- a flat curve expressing the law of the change in the sine of the angle depending on the change in the magnitude of the angle.

The construction of a sinusoid for a given circle diameter d is shown

in fig. 22.

To build it, divide the given circle into 12 equal parts; a segment equal to the length of a given circle (2pR) is divided into the same number of equal parts. Drawing horizontal and vertical straight lines through the division points, they find the sinusoid points at their intersection.

Spiral of Archimedes - e then a plane curve, described by a point, which rotates uniformly around a given center and at the same time uniformly moves away from it.

The construction of the Archimedes spiral for a given circle diameter D is shown in Fig.23.

The circumference and the radius of the circle are divided into 12 equal parts. Further construction is visible from the drawing.

When constructing conjugations and curved curves, one has to resort to the simplest geometric constructions - such as dividing a circle or a straight line into several equal parts, dividing an angle and a segment in half, building perpendiculars, bisectors, etc. All these constructions were studied in the discipline "Drawing" of the school course, therefore, they are not considered in detail in this manual.

1.5 Guidelines for implementation

PRACTICE #4

TOPIC: CONJUGATION OF LINES AND CIRCLES

JOINTS USED IN CONTOURS OF TECHNICAL DETAILS

A conjugation is a smooth transition from one line to another.

The point where one line meets another is called connection point.

Arcs, with the help of which a smooth transition from one line to another, is called conjugation arcs.

Tangent is called a straight line that has only one common point with a closed curve. This is the limiting position of the secant, the points of intersection of which with the curve, tending to each other, merge into one point - the point of contact.

The construction of conjugations is based on the properties of tangents to curves and is reduced to determining the position of the center of the conjugating arc and conjugation points (tangency), i.e. points at which the given lines pass into a mating arc

CORNER COMBINATION (INTERCECTING RIGHT COMBINATION)

Right angle mate

(Conjugation of intersecting lines at right angles)

In this example, we will consider the construction of a right angle mate with a given mate radius R. First of all, let's find the mate points. To find the junction points, you need to put a compass at the vertex of a right angle and draw an arc with a radius R until it intersects with the sides of the angle. The resulting points will be the conjugation points. Next, you need to find the center of pairing. The center of the mate will be a point equidistant from the sides of the corner. Let's draw two arcs from points a and b with a conjugation radius R until they intersect with each other. The point O obtained at the intersection will be the center of conjugation. Now, from the center of the junction of point O, we describe the arc with the junction radius R from point a to point b. The conjugation of the right angle is built.

Conjugation of an acute corner

(Conjugation of intersecting straight lines at an acute angle).

Another example of corner conjugation. In this example, an acute angle mate will be built. To construct a conjugation of an acute angle with a compass opening equal to the conjugation radius R, we draw two arcs from two arbitrary points on each side of the angle. Then we draw tangents to the arcs until they intersect at point O, the center of conjugation. From the resulting center of conjugation, we lower the perpendicular to each of the sides of the corner. This is how we get junction points a and b. Then we draw from the center of the pairing, points O, arc with fillet radius R, by connecting the junction points a and b. The conjugation of an acute angle is constructed.

Obtuse angle conjugation

(Conjugation of intersecting straight lines at an obtuse angle)

The conjugation of an obtuse angle is constructed by analogy with the conjugation of an acute angle. We also, first with a radius R, draw two arcs from two arbitrarily taken points on each side, and then draw tangents to these arcs until they intersect at point O, the center of the pair. Then we lower the perpendiculars from the center of the mate to each of the sides and connect with an arc equal to the radius of the mate of the obtuse angle R, received points a and b.